The above given Riemannian is built on the transformation space \(T_{CNN}\) . In fact just like the cases in both the quantum computation and the template matching, equivalently we can have another Riemannian manifold on the data space \(V_{CNN}\) . We can regard the CNN achieves a change of the metric from the input data space to the final output data space defined by \(\langle w,v\rangle _{in}=\langle U_*w,U_*v\rangle _{out}\) , where \(\langle \cdot ,\cdot \rangle _{in}\)  and \(\langle \cdot ,\cdot \rangle _{out}\)  are the metric on the input and output manifolds respectively and \(U_*\)  is the differential of the transformation \(U\) . So in this construction, the metric of the data space changes along the data flow in the network. As we indicated before, the input data can not be easily classified in the input data space by defining a simple distance between points, but it's much easier in the output data space even with an Euclidean distance. This means by applying the transformation \(U_{CNN}\) , the CNN transforms a highly curved input data space to a flatter output data space by a curvature flow like mechanism. This gives an intuitive picture on how a complex problem can be solved with the CNN by flattening a highly curved manifold as in [1]}. But it's difficult to estimate how the curvature of the input data space except for some special cases as in [2]}. In the rest of this paper we will focus on the Riemannian structure on \(T_{CNN}\)  since it's relatively easier to analysis the curvature of \(T_{CNN}\) .
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